arXiv subjects

Tridib Sadhu

Publications and source records attributed to Tridib Sadhu.

At least 37 records · Page 2Linked to original sources

arXiv2020

Large deviations in the symmetric simple exclusion process with slow boundaries

We obtain the exact large deviation functions of the density profile and of the current, in the non-equilibrium steady state of a one dimensional symmetric simple exclusion process coupled to boundary reservoirs with slow rates. Compared to earlier results, where rates at the boundaries are comparable to the bulk ones, we show how macroscopic fluctuations are modified when the boundary rates are slower by an order of inverse of the system length.

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arXiv2019

Large deviations conditioned on large deviations II: Fluctuating hydrodynamics

For diffusive many-particle systems such as the SSEP (symmetric simple exclusion process) or independent particles coupled with reservoirs at the boundaries, we analyze the density fluctuations conditioned on current integrated over a large time. We determine the conditioned large deviation function of density by a microscopic calculation. We then show that it can be expressed in terms of the solutions of Hamilton-Jacobi equations, which can be written for general diffusive systems using a fluctuating hydrodynamics description.

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arXiv2018

Large deviations conditioned on large deviations I: Markov chain and Langevin equation

We present a systematic analysis of stochastic processes conditioned on an empirical measure $Q_T$ defined in a time interval $[0,T]$ for large $T$. We build our analysis starting from a discrete time Markov chain. Results for a continuous time Markov process and Langevin dynamics are derived as limiting cases. We show how conditioning on a value of $Q_T$ modifies the dynamics. For a Langevin dynamics with weak noise, we introduce conditioned large deviations functions and calculate them using either a WKB method or a variational formulation. This allows us, in particular, to calculate the typical trajectory and the fluctuations around this optimal trajectory when conditioned on a certain value of $Q_T$.

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arXiv2018

Multiple singularities of the equilibrium free energy in a one-dimensional model of soft rods

There is a misconception, widely shared amongst physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at non-zero temperatures, can not show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counter-example. We consider thin rigid linear rods of equal length $2 \ell$ whose centers lie on a one-dimensional lattice, of lattice spacing $a$. The interaction between rods is a soft-core interaction, having a finite energy $U$ per overlap of rods. We show that the equilibrium free energy per rod $\mathcal{F}(\tfrac{\ell}{a}, β)$, at inverse temperature $β$, has an infinite number of singularities, as a function of $\tfrac{\ell}{a}$.

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arXiv2017

Generalized arcsine laws for fractional Brownian motion

The three arcsine laws for Brownian motion are a cornerstone of extreme-value statistics. For a Brownian $B_t$ starting from the origin, and evolving during time $T$, one considers the following three observables: (i) the duration $t_+$ the process is positive, (ii) the time $t_{\rm last}$ the process last visits the origin, and (iii) the time $t_{\rm max}$ when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the name of arcsine laws. We show how these laws change for fractional Brownian motion $X_t$, a non-Markovian Gaussian process indexed by the Hurst exponent $H$. It generalizes standard Brownian motion (i.e. $H=\tfrac{1}{2}$). We obtain the three probabilities using a perturbative expansion in $ε= H-\tfrac{1}{2}$. While all three probabilities are different, this distinction can only be made at second order in $ε$. Our results are confirmed to high precision by extensive numerical simulations.

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arXiv2017

Emergence and complexity in theoretical models of self-organized criticality

In this thesis we present few theoretical studies of the models of self-organized criticality. Following a brief introduction of self-organized criticality, we discuss three main problems. The first problem is about growing patterns formed in the abelian sandpile model (ASM). The patterns exhibit proportionate growth where different parts of the pattern grow in same rate, keeping the overall shape unchanged. This non-trivial property, often found in biological growth, has received increasing attention in recent years. In this thesis, we present a mathematical characterization of a large class of such patterns in terms of discrete holomorphic functions. In the second problem, we discuss a well known model of self-organized criticality introduced by Zhang in 1989. We present an exact analysis of the model and quantitatively explain an intriguing property known as the emergence of quasi-units. In the third problem, we introduce an operator algebra to determine the steady state of a class of stochastic sandpile models.

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arXiv2016

Correlations of the density and of the current in non-equilibrium diffusive systems

We use fluctuating hydrodynamics to analyze the dynamical properties in the non-equilibrium steady state of a diffusive system coupled with reservoirs. We derive the two-time correlations of the density and of the current in the hydrodynamic limit in terms of the diffusivity and the mobility. Within this hydrodynamic framework we discuss a generalization of the fluctuation dissipation relation in a non-equilibrium steady state where the response function is expressed in terms of the two-time correlations. We compare our results to an exact solution of the symmetric exclusion process. This exact solution also allows one to directly verify the fluctuating hydrodynamics equation.

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arXiv2015

Tagged particle in single-file diffusion

Single-file diffusion is a one-dimensional interacting infinite-particle system in which the order of particles never changes. An intriguing feature of single-file diffusion is that the mean-square displacement of a tagged particle exhibits an anomalously slow sub-diffusive growth. We study the full statistics of the displacement using a macroscopic fluctuation theory. For the simplest single-file system of impenetrable Brownian particles we compute the large deviation function and provide an independent verification using an exact solution based on the microscopic dynamics. For an arbitrary single-file system, we apply perturbation techniques and derive an explicit formula for the variance in terms of the transport coefficients. The same method also allows us to compute the fourth cumulant of the tagged particle displacement for the symmetric exclusion process.

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arXiv2015

Large deviation function of a tracer position in single file diffusion

Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.

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arXiv2015

Dynamical properties of single-file diffusion

We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory---the full single-time statistics, all multiple-time correlation functions, etc. are merely special cases.

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arXiv2014

Shared information in classical mean-field models

Universal scaling of entanglement estimators of critical quantum systems has drawn a lot of attention in the past. Recent studies indicate that similar universal properties can be found for bipartite information estimators of classical systems near phase transitions, opening a new direction in the study of critical phenomena. We explore this subject by studying the information estimators of classical spin chains with general mean-field interactions. In our explicit analysis of two different bipartite information estimators in the canonical ensemble we find that, away from criticality both the estimators remain finite in the thermodynamic limit. On the other hand, along the critical line there is a logarithmic divergence with increasing system-size. The coefficient of the logarithm is fully determined by the mean-field interaction and it is the same for the class of models we consider. The scaling function, however, depends on the details of each model. In addition, we study the information estimators in the micro-canonical ensemble, where they are shown to exhibit a different universal behavior. We verify our results using numerical calculations of two specific cases of the general Hamiltonian.

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arXiv2014

Melting of an Ising Quadrant

We consider an Ising ferromagnet endowed with zero-temperature spin-flip dynamics and examine the evolution of the Ising quadrant, namely the spin configuration when the minority phase initially occupies a quadrant while the majority phase occupies three remaining quadrants. The two phases are then always separated by a single interface which generically recedes into the minority phase in a self-similar diffusive manner. The area of the invaded region grows (on average) linearly with time and exhibits non-trivial fluctuations. We map the interface separating the two phases onto the one-dimensional symmetric simple exclusion process and utilize this isomorphism to compute basic cumulants of the area. First, we determine the variance via an exact microscopic analysis (the Bethe ansatz). Then we turn to a continuum treatment by recasting the underlying exclusion process into the framework of the macroscopic fluctuation theory. This provides a systematic way of analyzing the statistics of the invaded area and allows us to determine the asymptotic behaviors of the first four cumulants of the area.

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arXiv2014

Non-local response in a lattice gas under a shear drive

In equilibrium, the effect of a spatially localised perturbation is typically confined around the perturbed region. Quite contrary to this, in a non-equilibrium stationary state often the entire system is affected. This appears to be a generic feature of non-equilibrium. We study such non-local response in the stationary state of a lattice gas with a shear drive at the boundary which keeps the system out of equilibrium. We show that a perturbation in the form of a localised blockage at the boundary, induces algebraically decaying density and current profile. In two examples, non-interacting particles and particles with simple exclusion, we analytically derive the power-law tail of the profiles.

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arXiv2014

Large Deviations in Single File Diffusion

We apply macroscopic fluctuation theory to study the diffusion of a tracer in a one-dimensional interacting particle system with excluded mutual passage, known as single-file diffusion. In the case of Brownian point particles with hard-core repulsion, we derive the cumulant generating function of the tracer position and its large deviation function. In the general case of arbitrary inter-particle interactions, we express the variance of the tracer position in terms of the collective transport properties, viz. the diffusion coefficient and the mobility. Our analysis applies both for fluctuating (annealed) and fixed (quenched) initial configurations.

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arXiv2014

Long-range correlations in a locally driven exclusion process

We show that the presence of a driven bond in an otherwise diffusive lattice gas with simple exclusion interaction results in long-range density-density correlation in its stationary state. In dimensions $d>1$ we show that in the thermodynamic limit this correlation decays as $C(r,s)\sim (r^2+s^2)^{-d}$ at large distances $r$ and $s$ away from the drive with $|r-s|>>1$. This is derived using an electrostatic analogy whereby $C(r,s)$ is expressed as the potential due to a configuration of electrostatic charges distributed in $2d$-dimension. At bulk density $ρ=1/2$ we show that the potential is that of a localized quadrupolar charge. At other densities the same is correct in leading order in the strength of the drive and is argued numerically to be valid at higher orders.

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arXiv2013

A sandpile model for proportionate growth

An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. In this paper, we review our recent work on patterns formed by adding $N$ grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. We give several examples of such patterns. We discuss the special cases where the asymptotic pattern can be determined exactly. The effect of noise in the background or in the rules on the patterns is also discussed.

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arXiv2012

Modelling proportionate growth

An important question in biology is how the relative size of different organs is kept nearly constant during growth of an animal. This property, called proportionate growth, has received increased attention in recent years. We discuss our recent work on a simple model where this feature comes out quite naturally from local rules, without fine tuning any parameter. The patterns produced are composed of large distinguishable structures with sharp boundaries, all of which grow at the same rate, keeping their overall shapes unchanged.

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arXiv2012

An interface phase transition induced by a driven line in 2D

The effect of a localized drive on the steady state of an interface separating two phases in coexistence is studied. This is done using a spin conserving kinetic Ising model on a two dimensional lattice with cylindrical boundary conditions, where a drive is applied along a single ring on which the interface separating the two phases is centered. The drive is found to induce an interface spontaneous symmetry breaking whereby the magnetization of the driven ring becomes non-zero. The width of the interface becomes finite and its fluctuations around the driven ring are non-symmetric. The dynamical origin of these properties is analyzed in an adiabatic limit which allows the evaluation of the large deviation function of the driven-ring magnetization.

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